Extension of Uniformly Continuous Function into a Complete Metric Space

Let be a subset of , and be uniformly continuous, where is complete. Show that there exists a continuous function which maps the closure of to such that

So basically, we are trying to prove that a function can be extended to its domains closure, and still retain the same values on the original set.

It's late, so maybe I need to spend more time pondering this question, but I'm frankly a little stumped - of the continuity problems I have tackled so far from chapter 4 of Rudin, this is the first involving the hypothesis of a mapping into a complete metric space, so it suggests (at first to me anyway) a different method of proof.

I guess I would start off with first establishing that the closure of E is (obviously) a closed set, with a limit point in X (in fact, in E closure by definition). I can than approach this limit point with a convergent Cauchy sequence, and then the image of this sequence will again be Cauchy in Y and thus convergent in Y. These facts follow from the hypothesis that the function f is uniformly continuous, and Y is complete. Then I suppose I could somehow use the sequential characterization of continuity to conclude the theorem...but rigorizing this argument, and then actually proving that this "extension" big F is equal to little f on E, is something I'm not sure about.

Let be a subset of , and be uniformly continuous, where is complete. Show that there exists a continuous function which maps the closure of to such that

So basically, we are trying to prove that a function can be extended to its domains closure, and still retain the same values on the original set.

It's late, so maybe I need to spend more time pondering this question, but I'm frankly a little stumped - of the continuity problems I have tackled so far from chapter 4 of Rudin, this is the first involving the hypothesis of a mapping into a complete metric space, so it suggests (at first to me anyway) a different method of proof.

I guess I would start off with first establishing that the closure of E is (obviously) a closed set, with a limit point in X (in fact, in E closure by definition). I can than approach this limit point with a convergent Cauchy sequence, and then the image of this sequence will again be Cauchy in Y and thus convergent in Y. These facts follow from the hypothesis that the function f is uniformly continuous, and Y is complete. Then I suppose I could somehow use the sequential characterization of continuity to conclude the theorem...but rigorizing this argument, and then actually proving that this "extension" big F is equal to little f on E, is something I'm not sure about.

Anyway, any other ideas, or modifications to mine, for this problem?

Right. The basic idea is precisely as you stated. You can extend to by choosing, for each , a sequence in converging to (you know one exists by definition of closure for metric spaces) and then defining where the right hand side makes sense since uniformly continuous functions are Cauchy continuous, and since is completely we have that the desired limit exists.

From there you have to show that this mapping is well-defined, in the sense that it's independent of which sequence you choose that converges to a particular point. This isn't bad though. If then you know you can pick so large that they are both within ____ of and so by uniform continuity their images are with ____ of each other--ending in a for all and so they're equal. Note that this automatically gives you that (i.e. that extends ) since for each you can choose to be the constant sequence .

Showing its' continuous isn't bad then.

Does that help? If you're curious you can look on this blog post of mine, where I prove something (ever so) slightly more general.

Let me sleep on this tonight and write up a solution in the morning, with your extra thoughts in mind. Then I'll read your blog post afterward - it's a bit longer than I expected! On another note, I suppose it's time to boost my confidence a little more with this material - it has just been a literal crash course in "real" mathematics all quarter long for me; I'm a physics major, and so the only mathematics I've been exposed to prior to now has been largely computational.